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A Better Way to Teach Math
Is it possible to eliminate the bell curve in math class?
Imagine if someone at a dinner party casually announced, “I’m illiterate.” It would never happen, of course; the shame would be too great. But it’s not unusual to hear a successful adult say, “I can’t do math.” That’s because we think of math ability as something we’re born with, as if there’s a “math gene” that you either inherit or you don’t.
School experiences appear to bear this out. In every math class I’ve taken, there have been slow kids, average kids and whiz kids. It never occurred to me that this hierarchy might be avoidable. No doubt, math comes more easily to some people than to others. But the question is: Can we improve the methods we use to teach math in schools — so that everyone develops proficiency?
Looking at current math achievement levels in the United States, this goal might seem out of reach. But the experience of some educators in Canada and England, using a curriculum called Jump Math, suggests that we seriously underestimate the potential of most students and teachers.
“Almost every kid — and I mean virtually every kid — can learn math at a very high level, to the point where they could do university level math courses,” explains John Mighton, the founder of Jump Math, a nonprofit organization whose curriculum is in use in classrooms serving 65,000 children from grades one through eight, and by 20,000 children at home. “If you ask why that’s not happening, it’s because very early in school many kids get the idea that they’re not in the smart group, especially in math. We kind of force a choice on them: to decide that either they’re dumb or math is dumb.”
Children come into school with differences in background knowledge, confidence, ability to stay on task and, in the case of math, quickness. In school, those advantages can get multiplied rather than evened out. One reason, says Mighton, is that teaching methods are not aligned with what cognitive science tells us about the brain and how learning happens.
In particular, math teachers often fail to make sufficient allowances for the limitations of working memory and the fact that we all need extensive practice to gain mastery in just about anything. Children who struggle in math usually have difficulty remembering math facts, handling word problems and doing multi-step arithmetic (pdf). Despite the widespread support for “problem-based” or “discovery-based” learning, studies indicate that current teaching approaches underestimate the amount of explicit guidance, “scaffolding” and practice children need to consolidate new concepts. Asking children to make their own discoveries before they solidify the basics is like asking them to compose songs on guitar before they can form a C chord.
Mighton, who is also an award-winning playwright and author of a fascinating book called “The Myth of Ability,” developed Jump over more than a decade while working as a math tutor in Toronto, where he gained a reputation as a kind of math miracle worker. Many students were sent to him because they had severe learning disabilities (a number have gone on to do university-level math). Mighton found that to be effective he often had to break things down into minute steps and assess each student’s understanding at each micro-level before moving on.
Take the example of positive and negative integers, which confuse many kids. Given a seemingly straightforward question like, “What is -7 + 5?”, many will end up guessing. One way to break it down, explains Mighton, would be to say: “Imagine you’re playing a game for money and you lost seven dollars and gained five. Don’t give me a number. Just tell me: Is that a good day or a bad day?”
Separating this step from the calculation makes it easier for kids to understand what the numbers mean. Teachers tell me that when they begin using Jump they are surprised to discover that what they were teaching as one step may contain as many as seven micro steps. Breaking things down this finely allows a teacher to identify the specific point at which a student may need help. “No step is too small to ignore,” Mighton says. “Math is like a ladder. If you miss a step, sometimes you can’t go on. And then you start losing your confidence and then the hierarchies develop. It’s all interconnected.”
Mighton saw that if he approached teaching this way, he could virtually guarantee that every student would experience success. In turn, the children’s math anxiety diminished. As they grew more confident, they grew excited, and they began requesting harder challenges. “More than anything, kids love success,” he says, “and they love getting to higher levels, like in a video game.”
As the children experienced repeated success, it seemed to Mighton that their brains actually began to work more efficiently. Sometimes adding one more drop of knowledge led to a leap in understanding. One day, a child would be struggling; the next day she would solve a problem that was harder than anything she’d previously handled. Mighton saw that if you provided painstaking guidance, children would make their own discoveries. That’s why he calls his approach “guided discovery.”
The foundation of the process is building confidence, which Mighton believes should be the first goal of a math teacher. Confidence begets attention, which begets rich learning. “I’ve never met a teacher who will tell you that a student doesn’t need to be confident to excel in school,” explains Mighton. “But I’ve never seen a math curriculum that follows the implications of that idea rigorously.” Math is well-suited to build confidence. Teachers can reduce things to tiny steps, gauge the size of each step to the student and raise the bar incrementally.
When math is taught this way, surprising things happen.
Consider some of Jump’s results. It’s been used for four years in the public schools in Lambeth, one of the most economically depressed boroughs of London, England. Teachers placed into Jump the students who were struggling most in math. Among the 353 students who entered the program in fifth grade, only 12 percent began at grade level. Most were at least two grade levels behind and the vast majority were not expected to pass England’s grade six (KS2) national tests. But 60 percent did.
In rural Ontario, Jump was recently evaluated in a randomized controlled study involving 29 teachers and about 300 fifth-grade students (controlled studies of math programs are rare). Researchers from Toronto’s Hospital for Sick Children and the Ontario Institute for Studies in Education arranged for a control group of teachers to use their district’s standard curriculum while another group used Jump. Each set of teachers was given two days of training relevant to the materials they would be using.
In five months, researchers found substantial differences in learning. The Jump group achieved more than double the academic growth in core mathematical competencies evaluated using a well known set of standardized tests. (The study has not yet been published.) “Kids have to make pretty substantial gains in order to see this kind of difference,” explained Tracy Solomon, a developmental psychologist in the Research Institute at the Hospital for Sick Children who is the study’s lead author. “It’s impressive over a five-month period.”
Solomon believes that the key to Jump’s effectiveness is the way it “breaks math down to its component parts and builds it back up.” And she notes that this “flies in the face of the way math is typically taught.”
Interviewing teachers and principals, I have heard numerous stories of results like these. At times, they seem hard to reconcile with our assumptions about math. Isabel Grant, principal of the General Wolfe Elementary School, in Vancouver, British Columbia, has seen Jump produce impressive results in two schools where it has been used by a variety of teachers. Schools in British Columbia evaluate students based on whether they meet expectations for learning outcomes. “Teachers who used Jump were suddenly finding that they had all of their kids in the ‘fully meeting expectations’ category,” Grant told me. “It was such a foreign experience. It doesn’t typically happen when we’re teaching science or language arts. And they were kind of at a loss. ‘What do we do about this?’”
Another example is Mary Jane Moreau, who teaches at the Mabin School, an independent school in Toronto that does not screen students based on academic ability. Moreau, an experienced educator, dabbled with Jump for a year and started to see progress among her students, so she decided to immerse herself in the philosophy. “I was used to getting a bell curve in the past,” she told me, “but what I started seeing was all the kids getting between 90 and 100 percent on tests, and within months, they were all getting between 95 and 100 percent.”
She decided to see if the results would transfer to the standardized Test of Mathematical Abilities. Moreau teaches the same cohort of kids in fifth and sixth grades. Each September, for four years, the students wrote the test. From 2006 to 2007, the class percentile average jumped from 66 percent to 92 percent. From 2008 to 2009, with a new cohort, it increased from 54 percent to 98 percent.
Notably, the bell curve of the students’ scores shifted to the right and narrowed — which is to say that the performance differences between the “slow” kids and the “whiz” kids began to fade away. Moreau encouraged her sixth-grade students to enroll in the Mathematica Pythagoras contest, which attracts only five percent of Canadian students, most of whom would be deemed “gifted” in math. All but one did. For each group, 14 out of 17 students beat the contest average.
Moreau is a dedicated teacher — and she has the benefit of small classes — but, even so, she hadn’t seen results like this before. And it troubled her to think of students she had taught who didn’t have the opportunity to learn math this way. “When I think about what we’ve been doing for years when we could have been doing something else,” she told me, “I feel like I have to run so hard on this because I’m coming to the end of my career. But if I don’t help to change attitudes, I’ll feel like a criminal.”
Jump is a modest outfit. Mighton has a staff of 10 to create materials and conduct teacher trainings. With decisions about math curriculum highly politicized, it’s difficult for a small group to influence the debate. Big textbook companies and paid math consultants have a big say — and big investments — in what gets used.
It will take independent-minded educators to use Jump and see if its results can be replicated in more classrooms and schools. It’s hard to imagine what society might look like if we could undermine the math hierarchies that get solidified in grade school. These patterns tend to play out across society, often in negative ways. Wasn’t it the whiz kids who invented financial derivatives and subprime mortgages? And how many adults got themselves into hot water with their mortgages because, at bottom, they didn’t really understand the risks?
Even deeper, for children, math looms large; there’s something about doing well in math that makes kids feel they are smart in everything. In that sense, math can be a powerful tool to promote social justice. “When you have all the kids in a class succeeding in a subject, you see that they’re competing against the problem, not one another,” says Mighton. “It’s like they’re climbing a mountain together. You see a very healthy kind of competition. And it makes kids more generous to one another. Math can save us.”
On Friday, I’ll reply to comments, explain how Jump has helped one teacher to conquer her own math fears, and I’ll get into some more details about how the program works — including the vital role of bonus questions.